Isomorphisms in unital $C^*$-algebras

Abstract

It is shown that every almost linear bijection $h : A<br />rightarrow B$ of a unital $C^*$-algebra $A$ onto a unital<br />$C^*$-algebra $B$ is a $C^*$-algebra isomorphism when<br /> $h(3^n u y) = h(3^n u) h(y)$ for all<br />unitaries $u in A$, all $y in A$, and all $nin mathbb Z$, and<br />that almost linear continuous bijection $h : A rightarrow B$ of a<br />unital $C^*$-algebra $A$ of real rank zero onto a unital<br />$C^*$-algebra $B$ is a $C^*$-algebra isomorphism when $h(3^n u y) =<br />h(3^n u) h(y)$ for all<br /> $u in { v in A mid v = v^*, |v|=1, v text{ is invertible} }$, all<br />$y in A$, and all $nin mathbb Z$.<br /><br />Assume that $X$ and $Y$ are left normed modules over a unital<br />$C^*$-algebra $A$. It is shown that every surjective isometry $T : X<br />rightarrow Y$, satisfying $T(0) =0$ and $T(ux) = u T(x)$ for all $x<br />in X$ and all unitaries $u in A$, is an $A$-linear isomorphism.<br />This is applied to investigate $C^*$-algebra isomorphisms in unital<br />$C^*$-algebras.